Finite Group Databasebeta

← #10826915

Subgroups — #10826915

Order 768.

1682 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic, complemented
2#87251C211yesmaximal, characteristic, complemented
3#84081C212yesmaximal, complemented
4C2×#5095C211yesmaximal, characteristic, complemented
5C2×#5062C211yesmaximal, characteristic, complemented
6#72133C211yesmaximal, characteristic, complemented
7#71795C212yesmaximal, complemented
8#71784C211yesmaximal, characteristic, complemented
9#71730C212yesmaximal, complemented
10#71724C211yesmaximal, characteristic, complemented
11#71591C212yesmaximal, complemented
12#23917—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C2×#1007C2211yescharacteristic
16C3×#536—24no—
17C3×#536—24no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C2×C3×#124C2211yescharacteristic
22C2×C3×#123—22no—
23C2×C3×C22⋊Q8—22no—
24C2×C3×#117C2211yescharacteristic
25#5817—24no—
26#5816—22no—
27#5816C2212yes—
28Q8×#67—24no—
29#5812C2211yescharacteristic
30C2×#942C2212yes—
31C2×#932C2212yes—
32#5791—22no—
33#5791C2212yes—
34#5239—24no—
35#5238—24no—
36C2×#836C2211yescharacteristic
37C2×#836C2211yescharacteristic
38C2×#836C2212yes—
39#5225C2212yes—
40#5215C2212yes—
41#5207C2212yes—
42#5095—24no—
43#5062—24no—
44C2×#805C2212yes—
45C2×#805C2212yes—
46#4523C2212yes—
47#4523C2212yes—
48#3486—31no—
49C2×#536D311yescharacteristic, complemented
50C2×#456—31no—
51C2×#454—31no—
52#3005—32no—
53#2005—32no—
54#2003—32no—
55#1897—32no—
56#1888—31no—
57#1886—31no—
58#1784—31no—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—24no—
62C2×C3×#41—42no—
63C22×C3×Q8—22no—
64C22×C3×Q8—22no—
65C22×C3×Q8—24no—
66C22×C3×Q8—42no—
67C22×C3×Q8D412yes—
68C22×C3×Q8C2311yescharacteristic
69C22×C3×Q8C2311yescharacteristic
70C22×C3×Q8C2311yescharacteristic
71C22×C3×D4—21no—
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4—21no—
74C23×C3×C4D412yes—
75C23×C3×C4C2311yescharacteristic
76#1007—22no—
77#1007—22no—
78#1007—42no—
79C2×#239—22no—
80C2×D3×Q8—22no—
81C2×#235—22no—
82C2×#235C2312yes—
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8—22no—
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8D414yescomplemented
87C3×#124—22no—
88C3×#124—22no—
89C3×#124—22no—
90C3×#124D414yescomplemented
91C3×#123—216no—
92C3×C22⋊Q8—216no—
93C3×#117—22no—
94C3×#117—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103#953—22no—
104#947—24no—
105#947—22no—
106#946—28no—
107#946—24no—
108Q8×Dic3—24no—
109#944—24no—
110#944—22no—
111C2×#216—22no—
112#942—24no—
113#942—42no—
114C2×#215—22no—
115C2×#214—22no—
116#932—24no—
117#932—42no—
118C2×#213—22no—
119#930—24no—
120#930D414yes—
121#929—24no—
122#929—22no—
123C4×#67—28no—
124C4×#67—24no—
125#926—22no—
126#926C2312yes—
127#924—24no—
128#924D414yes—
129C2×#208C2312yes—
130C2×#208C2312yes—
131#836—22no—
132#836—22no—
133#836—42no—
134#835—24no—
135#832—24no—
136#830—24no—
137#805—24no—
138#805—24no—
139C2×2−1+4—31no—
140C22×#41C2×D311yescharacteristic
141C23×Q8C2×D311yescharacteristic
142C2×#137—32no—
143C2×AGL(1,ℤ/8)—32no—
144D4×Q8—64no—
145#548—64no—
146#536—24no—
147#536—24no—
148#536—32no—
149#536—62no—
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×C4⋊Q8C2×D311yescharacteristic
152C2×#124C2×D311yescharacteristic
153C2×#124C2×D311yescharacteristic
154#525—32no—
155#525—62no—
156C2×#123—22no—
157C2×C22⋊Q8—22no—
158#519—31no—
159C2×#117C2×D311yescharacteristic
160#456—64no—
161#454—64no—
162#452—64no—
163#449—64no—
164C2×#104—32no—
165C2×#104—32no—
166#419—32no—
167#418—32no—
168C2×#101—31no—
169C2×#101—31no—
170C2×#101—32no—
171#407—32no—
172#339—32no—
173#339—32no—
174C24×C3—21no—
175C24×C3C2×D411yescharacteristic
176C3×#41—48no—
177C3×#41—48no—
178C2×C3×Q8—24no—
179C2×C3×Q8—24no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—22no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8—22no—
189C2×C3×Q8—42no—
190C2×C3×Q8—48no—
191C2×C3×Q8—48no—
192C2×C3×Q8C2×D412yes—
193C2×C3×Q8C2×D412yes—
194C2×C3×Q8C2×D412yes—
195C2×C3×D4—24no—
196C2×C3×D4—22no—
197C2×C3×D4—21no—
198C2×C3×D4—22no—
199C2×C3×D4—42no—
200C2×C3×D4—42no—
201C2×C3×D4C2×D412yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—22no—
205C22×C3×C4—24no—
206C22×C3×C4—24no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—24no—
211C22×C3×C4—42no—
212C22×C3×C4—42no—
213C22×C3×C4—42no—
214C22×C3×C4C2×D414yes—
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C2×#67—24no—
220C2×#67—22no—
221#239—44no—
222#239—44no—
223D3×Q8—44no—
224D3×Q8—44no—
225#235—22no—
226#235—22no—
227#235—22no—
228#235—22no—
229#235—44no—
230#235—44no—
231C2×D12—22no—
232C2×D12C2×D412yes—
233C2×C4×D3—24no—
234C2×C4×D3—22no—
235C2×Dic6—22no—
236C2×Dic6C2×D412yes—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—28no—
239C3×C4⋊C4—28no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—28no—
242C3×C22⋊C4—28no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42C2×D412yes—
246C3×C42C2×D412yes—
247#217—24no—
248#217—22no—
249#216—44no—
250#215—44no—
251#214—44no—
252#213—44no—
253(C2×D3)⋊C4—28no—
254(C2×D3)⋊C4—24no—
255C4⋊Dic3—24no—
256C4⋊Dic3C2×D414yes—
257Dic3⋊C4—28no—
258Dic3⋊C4—24no—
259C4×Dic3—24no—
260C4×Dic3—22no—
261#208—22no—
262#208—22no—
263#208—22no—
264#208—22no—
265C2×C3⋊C8—22no—
266C2×C3⋊C8—22no—
2672−1+4—62no—
2682−1+4—62no—
2692−1+4—122no—
270C2×#41—22no—
271C2×#41—22no—
272C2×#41—24no—
273C2×#41—32no—
274C2×#41—42no—
275C2×#41—62no—
276C2×#41—62no—
277C22×Q8—22no—
278C22×Q8—22no—
279C22×Q8—24no—
280C22×Q8—42no—
281C22×Q8—62no—
282C22×Q8#6712yes—
283C22×Q8C22×D311yescharacteristic
284C22×Q8C22×D311yescharacteristic
285C22×Q8C22×D311yescharacteristic
286C22×D4—21no—
287C22×D4C22×D311yescharacteristic
288C23×C4—21no—
289C23×C4#6712yes—
290C23×C4C22×D311yescharacteristic
291#137—64no—
292#137—122no—
293AGL(1,ℤ/8)—64no—
294AGL(1,ℤ/8)—122no—
295C2×Q16—62no—
296C2×SD16—62no—
297C2×SD16—62no—
298C2×D8—62no—
299C2×M16—32no—
300C2×M16—32no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—62no—
305C4⋊Q8—64no—
306C4⋊Q8#6714yescomplemented
307#124—22no—
308#124—22no—
309#124—22no—
310#124—62no—
311#124—64no—
312#124#6714yescomplemented
313#123—216no—
314#123—62no—
315#123—64no—
316C22⋊Q8—216no—
317C22⋊Q8—34no—
318C22⋊Q8—64no—
319C22⋊Q8—64no—
320C22⋊Q8—68no—
321#121—34no—
322#121—64no—
323#120—62no—
324C4×Q8—64no—
325C4×D4—68no—
326C4×D4—64no—
327#117—22no—
328#117—22no—
329#117—32no—
330#117—62no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C42C22×D311yescharacteristic
338C2×C42C22×D311yescharacteristic
339#104—64no—
340#104—64no—
341#103—64no—
342#102—64no—
343#101—62no—
344#101—62no—
345#101—122no—
346C22⋊C8—64no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—22no—
350C23×C3—42no—
351C23×C3—44no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3#12012yes—
355C23×C3C22×D411yescharacteristic
356C22×D3—22no—
357C3×Q8—24no—
358C3×Q8—24no—
359C3×Q8—24no—
360C3×Q8—48no—
361C3×Q8—42no—
362C3×Q8—42no—
363C3×Q8—42no—
364C3×D4—24no—
365C3×D4—48no—
366C3×D4—42no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—24no—
372C2×C3×C4—24no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—48no—
380C2×C3×C4—48no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4#12012yes—
386C2×C3×C4#12014yes—
387C2×C3×C4C22×D412yes—
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4H(5,𝔽2)12yes—
391#67—48no—
392#67—42no—
393#67—42no—
394C2×Dic3—24no—
395C2×Dic3—22no—
396D12—22no—
397D12—22no—
398D12—44no—
399C4×D3—48no—
400C4×D3—42no—
401C4×D3—42no—
402Dic6—22no—
403Dic6—22no—
404Dic6—44no—
405C3⋊C8—42no—
406C3⋊C8—42no—
407C24—21no—
408C24C2×#6711yescharacteristic
409#41—48no—
410#41—48no—
411#41—62no—
412#41—62no—
413#41—62no—
414#41—62no—
415#41—124no—
416#41—124no—
417#41—124no—
418#41—124no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—32no—
431C2×Q8—42no—
432C2×Q8—48no—
433C2×Q8—48no—
434C2×Q8—62no—
435C2×Q8—124no—
436C2×Q8—124no—
437C2×Q8C2×#6712yes—
438C2×Q8C2×#6712yes—
439C2×Q8C2×#6712yes—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—32no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—62no—
448C2×D4—64no—
449C2×D4—62no—
450C2×D4C2×#6712yes—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—64no—
464C22×C4—62no—
465C22×C4D3×D414yes—
466C22×C4C2×#6712yes—
467C22×C4C2×#6711yescharacteristic
468C22×C4C2×#6712yes—
469C22×C4C23×D311yescharacteristic
470Q16—124no—
471SD16—124no—
472SD16—124no—
473D8—124no—
474M16—62no—
475M16—62no—
476M16—62no—
477M16—62no—
478C2×C8—62no—
479C2×C8—62no—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—34no—
484C4⋊C4—64no—
485C4⋊C4—68no—
486C4⋊C4—64no—
487C22⋊C4—24no—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—68no—
491C22⋊C4—64no—
492C22⋊C4—62no—
493C22⋊C4—64no—
494C42—21no—
495C42—21no—
496C42—64no—
497C42—62no—
498C42C2×#6712yes—
499C42C2×#6712yes—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—44no—
507C22×C3—42no—
508C22×C3—82no—
509C22×C3—82no—
510C22×C3C2×#12011yescharacteristic
511C22×C3C2×#12011yescharacteristic
512C22×C3#53411yescharacteristic
513C2×D3—42no—
514C2×D3—44no—
515C3×C4—24no—
516C3×C4—22no—
517C3×C4—42no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—42no—
521C3×C4D4212yes—
522C3×C4D4212yes—
523Dic3—42no—
524Dic3—44no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—42no—
529C23—44no—
530C23—41no—
531C23—41no—
532C23—62no—
533C23#93712yes—
534C23C22×#6711yescharacteristic
535Q8—24no—
536Q8—24no—
537Q8—24no—
538Q8—48no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—62no—
543Q8—62no—
544Q8—124no—
545D4—24no—
546D4—48no—
547D4—42no—
548D4—62no—
549D4—62no—
550D4—124no—
551D4—128no—
552D4—122no—
553D4—122no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—64no—
573C2×C4—62no—
574C2×C4—128no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4#93712yes—
578C2×C4#95314yes—
579C2×C4C2×D3×D412yes—
580C2×C4#100412yes—
581C2×C4C22×#6711yescharacteristic
582C2×C4C22×#6711yescharacteristic
583C8—122no—
584C8—122no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—42no—
588C2×C3—82no—
589C2×C3#240311yescharacteristic
590C2×C3#302012yes—
591D3—82no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—42no—
600C22—82no—
601C22—82no—
602C22—122no—
603C22—124no—
604C22C2×#93711yescharacteristic
605C22#580411yescharacteristic
606C22C2×#95311yescharacteristic
607C4—22no—
608C4—24no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—122no—
614C4—124no—
615C4D4×#6712yes—
616C4D4×#6712yes—
617C3#2391711yescharacteristic, complemented, Sylow
618C2—21no—
619C2—21no—
620C2—42no—
621C2—82no—
622C2—242no—
623C2#8341512yes—
624C2#8405211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#536C2
socleC22×C3C2×#120
hypercenterC2×#536D3
hypocenterC3#23917
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#536

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C3×#536C2semidirect
2. #87251C2semidirect
3. #84081C2semidirect
4. C2×#5095C2semidirect
5. C2×#5062C2semidirect
6. #72133C2semidirect
7. #71795C2semidirect
8. #71784C2semidirect
9. #71730C2semidirect
10. #71724C2semidirect
11. #71591C2semidirect
49. C2×#536D3semidirect
86. C3×C4⋊Q8D4semidirect
90. C3×#124D4semidirect
306. C4⋊Q8#67semidirect
312. #124#67semidirect
617. C3#23917semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#5362
#872512
#840812
C2×#50952
C2×#50622
#721332
#717952
#717842
#717302
#717242
#715912
#239173